Nonexistence of Simple Graphs with Laplacian Spectrum {0,1,...,n-1} for n=16 and 20: Exact Computer-Assisted Proofs

We present exact computer-assisted nonexistence proofs for the order-16 and order-20 cases of the S_n,n conjecture. No simple graph on 16 vertices has Laplacian spectrum {0,1,...,15}, and no simple graph on 20 vertices has Laplacian spectrum {0,1,...,19}. The proofs combine exact enumeration, spectral and graph-theoretic relaxations, Farkas certificates and exhaustive integer branching where required. The deposit includes both certificate archives, exact verifiers, coverage records and replay logs. These finite-order results do not settle the conjecture in general. This is a non-peer-reviewed preprint.